Arb: a C library for ball arithmetic

Arb 1 is a new open source C library for provably correct arbitrary-precision numerics, extending FLINT [3] (which provides fast arithmetic over various exact rings) to the real and complex numbers. Following the example of iRRAM [7] and Mathemagix [13], Arb performs automatic error propagation using ball arithmetic [12] (not to be confused with heuristic significance arithmetic as used e.g. in Mathematica [9]). This gives performance close to floating-point arithmetic such as provided by MPFR [2] while avoiding the cost at high precision of endpoint-based interval arithmetic as provided for instance by MPFI [8]. One of our motivations for developing a new library has been to provide a low-level, low-overhead interface, and our implementation differs from others in some technical aspects. Arb also provides fast polynomial arithmetic, to our knowledge only available in Mathemagix and without error control in MPFRCX [1], as well as matrix arithmetic. Finally, Arb implements some special functions that have been absent from arbitrary-precision interval software, with performance that compares favorably to available nonrigorous implementations. The presentation covers implementation details and shows some benchmarks.